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Interactions of spectator partons in the Drell-Yan process
Volume 117B, number 1, 2
PHYSICS LETTERS
4 November 1982
INTERACTIONS OF SPECTATOR PARTONS IN THE DRELL-YAN PROCESS W.W. LINDSAY, D.A. ROSS and C.T. SACHRAJDA Department o f Physics, University of Southampton, Southampton S 0 9 5NH, UK
Received 12 May 1982 Revised manuscript received 1 September 1982
We report on a calculation of all two-loop graphs in QCD involving the interactions of virtual gluons with the spectator quark in the process quark + hadron ~ lepton pair + anything. Because of the different ie's in the propagators there are contributions from individual diagrams which differ from the corresponding contributions in deep inelastic scattering. We find however that the sum of all these non-factorising terms is zero.
In 1970 Drell and Yan [1] suggested that at high energies the process hadron + hadron -+ massive lepton pai:r + anything occurs by the annihilation o f a quark from one o f the initial hadrons with an antiquark from the other. Since quark and antiquark distributions are measured in deep inelastic lepton hadron scattering experiments, in their model one is able to make absolute predictions for do/dQ 2, where Q is the invariant mass o f the lepton pair. With the advent ofperturbative QCD and the theorem on the factorization o f mass singularities several years later, it was seen that the D r e l l - Y a n prediction was indeed asymptotically correct in QCD and that there are corrections to it o f O(as(Q2)) which are calculable and have been calculated [2]. These corrections are large and appear to be required by the data [3], which might seem to be a satisfactory situation. However, during the last two years there have arisen two theoretical problems which have shed some doubt on the validity o f the QCD predictions for the D r e l l Yan process. These two problems are (i) the non-cancellation o f infrared divergences in non-abelian gauge theories [4] and (ii) the importance o f the "initial state interactions" which Bodwin et al. claim do invalidate the D r e l l - Y a n prediction. It seems that the non-cancellation o f infrared divergences only occurs at the nonleading twist level, (i.e. is suppressed by powers o f Q2) although a p r o o f o f this to all orders does not yet exist, so (i) may not be a problem. In this letter we are concerned with the second problem, and will present the results o f a systematic study to two loops o f all dia0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
grams with virtual gluons which contribute to the process quark + hadron ~ lepton pair + anything. We have worked throughout in the F e y n m a n gauge. We find that there are such contributions from individual diagrams, but their sum is zero; this differs from the conclusion of Bodwin et al. [5]. The fundamental question we are interested in is whether the QCD predictions, which are calculated using on-shell quarks and gluons and neglecting interactions of "spectator" quarks are valid in the physically relevant case of hadronic beams and targets. We start our investigation by studying a simple one-loop graph involving the interaction o f a spectator quark, which contributes to the deep inelastic structure function of a hadron (fig. 1 a). We will use Sudakov variables and to this end we define q' = q + x p ,
(1)
and p' = p - (p2/s) q' ,
(2)
where s = 2 p q and x is the Bjorken scaling variable (x = - q 2 / 2 p q ) . The Sudakov variables {c~, t3, k±} are then defined by k = a p ' + (3q' + k z .
(3)
Simple power counting then shows that the following (overlapping) regions of phase space give a scaling (leading power) contribution: (a) The "collinear" region: a ~ O(1),/3 ~ O(k2/s), 105
Volume 117B, number 1, 2
PHYSICS LETTERS
k± fixed (s ~ oo). The contributions from this region have already been studied to some extent [6], with the result that at least in low orders o f perturbation theory, they do not spoil the Drell-Yan picture. (b) The "Glauber" region: a ~ 13 ~ k2 /s, k± fixed (s -+ ~). It has recently been pointed out that this is an important region of phase space [5], moreover that contributions from two-loop graphs invalidate the usual Drell-Yan predictions. Below we shall study this in more detail. (c) The "infrared" region: a ~/3 ~ k ± / ~ , k± small (of the order of the infrared cut-off). Up to two years ago it was hoped that infrared divergences cancel in QCD in a way similar to the Bloch-Nordsieck mechanism in QED. We know now that this is not true, however it seems that the non-cancelling terms are suppressed by powers o f s [4] (or Q2). In this paper we shall be interested in the contributions from regions (a) and (b); specifically we shall integrate over all a and/3 keeping k± fixed. In order to be able to do explicit calculations we have chosen to work with massless scalar quarks, which are charged and coloured and described by the field X, and a neutral, colour singlet scalar hadron described by the field ¢. The hadronic wave function is then assumed to come from a XCX+Xinteraction term. X has the dimension of mass which has the effect that the wavefunction is soft, a physically desirable feature. Since we are dealing with massless quarks we have taken the hadron momentum, p, to be spacelike in order to guarantee the stability of the wavefunction. With this model we are able to study the problem of spectator quark interactions in detail, however since the results presented below seem to depend primarily on the analytic structure of the graphs (the ie terms in the propagators) we feel confident that they are more general than the model *l. The Sudakov variables for the outgoing quark momentum, P2, are
4 November 1982
F IDI(X ) = is 2 J -
*i It is for this reason that we do not worry about the unphysical nature of the spacelike hadron momentum. 106
k 2 + i e ) [ ( 1 - a ) ( p 2 - / 3 s ) - k 2 +ie]
X [(c~-x)(1 + / 3 ) s - k 2 +ie] -1 .
1 (5)
The factor i comes from the usual Feynman rules and the powers o f s come from the jacobian (d4k = ~s da × dfl d2kz) and the numerator (we have now changed variables k ~ P l - q - k from the labelling of fig. la). Now we consider the equivalent contribution from the graph shown in fig. lb to the Drell-Yan process quark + hadron -+ lepton pair + anything. This graph is very similar to that of fig. la and the integral to be calculated is
/By(O=-is2f (a~3s X [(1 - a -
da d/3 d2k± k 2 + ie)[(1 - a ) ( p 2 _/30 - k2 + ie]
o)(p 2 - p s - / 3 s ) -
× [(1 - o - a ) ( 1 - / 3 - n ) s - ( p
(P_L+kz)2 + i e ] - 1 a +k±) 2 +ie] -1 . (6)
The dominant contribution to these integrals comes from the small/3 region and it is a simple matter to show that IDI(r)
IDy(r) d2k±
= _4zr2i
f (Pl+k±)Z[k2-r(
1 - r)p2l
(7)
Thus the difference of the deep-inelastic and DrellYan amplitudes is purely imaginary. Since the contributions to the structure functions and Drell-Yan cross sections are equal to twice the real part of the interferences of the one-loop graphs of fig. 1 (together with the corresponding graphs involving "seagull" vertices) with the tree diagrams, we obtain no non-factorizing q
l
(4)
(or x +-+ r in the Drell-Yan case). After combining with the corresponding diagram involving a "seagull" vertex the integral to be evaluated is
(afls
X [(1 -- a - o)(p 2 - ; s --/30 -- (P± + k±) 2 + iel
P2 = ( o , p , p i } ((1 - x), p2/(1 - x ) s, P i }
da dfl d2k±
f
(o)
0 k,
P
~-
(b}
Fig. 1. One-loop graphs involving spectator interactions which contribute to (a) the structure function and (b) the Drell-Yan cross section quark + hadron -~ lepton pair + anything.
Volume 117B, number 1, 2
PHYSICS LETTERS
contribution from these one-loop graphs. Thus the Drell-Yan picture is not spoilt by '"collinear" or "Glauber" region contributions to one-loop graphs involving spectator interactions. On closer examination we find that the real parts of the two integrals in (5) and (6) are equal, whereas the imaginary parts have the opposite sign (but the same magnitude). The opposite sign comes from the ie's in the last propagator in (5) and (6), if we switch the sign of the ie in one of the two integrals then the imaginary parts become equal. We now come to the two-loop graphs. Bodwin et al., working in an axial gauge [5], suggest that the graph of fig. 2b has a contribution from the "Glauber" region which spoils the Drell-Yan factorization. We find however that (at least in the Feynman gauge) there is a precisely compensating contribution due to that of fig. 2a, which contributes to the structure function. The difference of the two corresponding integrals is again purely imaginary. We find that this happens for all the contributions except four! The first exception is the interference of the graphs of figs. 3a,b with those of figs. la,b, respectively. In the Drell-Yan case there is a contribution where we keep the imaginary parts of both figs. lb and 3b, with no corresponding contribution in the deep inelastic case since the diagram of fig. 3a is purely real (q2 < 0). The other three exceptions are the contributions of figs. 4, 5 and 6. In each case we Fred that the difference between the deep inelastic and Drell-Yan diagrams has a real part and has the correct scaling behaviour. Each of the four contributions is of the form ( g 4 / s ) ( 1 --
r){k2,k2s[r(1
× [¢(1 - ~ ) p 2 _ p ? ] - i
(o)
- r ) p 2 _ (p± + kl±)2 ] }-1 ,
(8)
4 November 1982
/ (o)
(b)
Fig. 3. Some one-loop graphs whose interference with those of fig. 1 gives a non-factorizing contribution for the Drell-Yan cross section (see text).
'k' {a)
{hi
Fig. 4. Two-loop graphs contributing to (a) the structure function and (b) the Drell-Yan cross section which are not related by the Drell Yan formula. with relative weights '
'
'
-- ~CFCA) , --7~CFCA , - ~ C F C A , from figs. 3, 4, 5 and 6, respectively. C F and C A are the eigenvahies of the quadratic Casimir operator in the fundamental and adjoint representations, respectively. All other contributions vanish when the deep inelastic contribution is compared to the Drell-Yan one and hence there are no nonfactorizing terms to this order!!! For completeness we would like to point out some of the things we have not yet calculated:
(b)
Fig. 2. Two-loop graphs contributing to (a) the structure function and (b) the DreU-Yan cross section which are related by the Drell-Yan formula (in the Feynman gauge).
(Q)
(N
Fig. 5. Two-loop graphs which are not related by the DreUYan formula. 107
Volume 117B, number 1, 2
(a)
PHYSICS LETTERS
(b)
Fig. 6. Two-loop graphs which are not related by the DrellYan formula. (a) We have not calculated the contribution from diagrams with real gluons. (b) We have restricted our transverse momenta to be much smaller than s and much larger than any infrared cut-off. We would like to extend the calculation to include all transverse momenta, however this is rather complicated at the two-loop level. (c) The higher order calculations in QCD [2] are performed with on-shell quarks and gluons. Perhaps the same calculations but now performed with one quark slightly off-shell will give a different result. At first sight this might be considered to be an attractive possibility since, for example, it is known that contributions from diagrams related to the Sudakov form factor depend on whether the external legs are on-shell or not. Landshoff and Stifling [7] have recently shown that this is precisely what happens in oneloop graphs when one calculates the K factor o f the D r e l l - Y a n process in h a d r o n - h a d r o n scattering. One can either calculate this factor with on-shell quarks and gluons and neglect spectator interactions (as is usually done) or one can calculate with both incoming partons off-shell, but then one has to include the spectator interactions. Both procedures give the same result. They find that the difference between the off-shell and onshell contributions from individual diagrams comes from the region o f small transverse m o m e n t u m (the "infrared" region) which is why we do not expect any
108
4 November 1982
real terms on the right hand side o f eq. (7) when considering only the "collinear" and "Glauber" regions. In the same way we would expect that at the two-loop level any contributions from diagrams not involving spectator interactions, which depend on whether the active quarks are on-shell or not would arise from the "infrared" regions and not from the phase space regions considered in this letter. However this still has to be checked. The results of this paper are reassuring in that we have not found any long distance contributions which cannot be absorbed into the parton distribution functions. It is now important to determine whether this is true to all orders, and if so, to understand why it is so in a more physical way, rather than relying on an apparently "miraculous" cancellation of terms. References
[1] S.D. DreU and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316; Ann. Phys. 66 (1971) 578. [2] G. Altarelii, R.K. Ellis and G. Martinelli, Nucl. Phys. B143 (1978) 521; B146 (1978) 544 (E); B157 (1979) 461; J. Kubar-Andr6 and F.E. Paige, Phys. Rev. D19 (1979) 221; B. Humpert and W.L. van Neerven, Phys. Lett. 84B (1979) 327;85B (1979) 293. [3] J. Badier et al., Phys. Lett. 89B (1979) 145; 96B (1980) 422; M.J. Corden et al., Phys. Lett. 96B (1980) 417. [4] R. Doria, J. Frenkel and J.C. Taylor, Nucl. Phys. B168 (1980) 93; C. Di'Lieto, S. Gendron, I.G. Halliday and C.T. Sachrajda, Nucl. Phys. B183 (1981) 223. [5] G.T. Bodwin, S.J. Brodsky and G.P. Lepage, SLAC preprint SLAC-PUB-2787 (1981). [6] C.T. Sachrajda and S. Yankielowicz, unpublished, reported in: J. Ellis and C.T. Sachrajda, Quantum chromodynamics and its applications in quarks and leptons, Proc. 1979 Cargbse Summer Institute, eds. M. L6vy et al. (Plenum, New York, 1980). [7] P.V. Landshoff and W.J. Stirling, Cambridge University preprint DAMTP 82/3 (1982).
PHYSICS LETTERS
4 November 1982
INTERACTIONS OF SPECTATOR PARTONS IN THE DRELL-YAN PROCESS W.W. LINDSAY, D.A. ROSS and C.T. SACHRAJDA Department o f Physics, University of Southampton, Southampton S 0 9 5NH, UK
Received 12 May 1982 Revised manuscript received 1 September 1982
We report on a calculation of all two-loop graphs in QCD involving the interactions of virtual gluons with the spectator quark in the process quark + hadron ~ lepton pair + anything. Because of the different ie's in the propagators there are contributions from individual diagrams which differ from the corresponding contributions in deep inelastic scattering. We find however that the sum of all these non-factorising terms is zero.
In 1970 Drell and Yan [1] suggested that at high energies the process hadron + hadron -+ massive lepton pai:r + anything occurs by the annihilation o f a quark from one o f the initial hadrons with an antiquark from the other. Since quark and antiquark distributions are measured in deep inelastic lepton hadron scattering experiments, in their model one is able to make absolute predictions for do/dQ 2, where Q is the invariant mass o f the lepton pair. With the advent ofperturbative QCD and the theorem on the factorization o f mass singularities several years later, it was seen that the D r e l l - Y a n prediction was indeed asymptotically correct in QCD and that there are corrections to it o f O(as(Q2)) which are calculable and have been calculated [2]. These corrections are large and appear to be required by the data [3], which might seem to be a satisfactory situation. However, during the last two years there have arisen two theoretical problems which have shed some doubt on the validity o f the QCD predictions for the D r e l l Yan process. These two problems are (i) the non-cancellation o f infrared divergences in non-abelian gauge theories [4] and (ii) the importance o f the "initial state interactions" which Bodwin et al. claim do invalidate the D r e l l - Y a n prediction. It seems that the non-cancellation o f infrared divergences only occurs at the nonleading twist level, (i.e. is suppressed by powers o f Q2) although a p r o o f o f this to all orders does not yet exist, so (i) may not be a problem. In this letter we are concerned with the second problem, and will present the results o f a systematic study to two loops o f all dia0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
grams with virtual gluons which contribute to the process quark + hadron ~ lepton pair + anything. We have worked throughout in the F e y n m a n gauge. We find that there are such contributions from individual diagrams, but their sum is zero; this differs from the conclusion of Bodwin et al. [5]. The fundamental question we are interested in is whether the QCD predictions, which are calculated using on-shell quarks and gluons and neglecting interactions of "spectator" quarks are valid in the physically relevant case of hadronic beams and targets. We start our investigation by studying a simple one-loop graph involving the interaction o f a spectator quark, which contributes to the deep inelastic structure function of a hadron (fig. 1 a). We will use Sudakov variables and to this end we define q' = q + x p ,
(1)
and p' = p - (p2/s) q' ,
(2)
where s = 2 p q and x is the Bjorken scaling variable (x = - q 2 / 2 p q ) . The Sudakov variables {c~, t3, k±} are then defined by k = a p ' + (3q' + k z .
(3)
Simple power counting then shows that the following (overlapping) regions of phase space give a scaling (leading power) contribution: (a) The "collinear" region: a ~ O(1),/3 ~ O(k2/s), 105
Volume 117B, number 1, 2
PHYSICS LETTERS
k± fixed (s ~ oo). The contributions from this region have already been studied to some extent [6], with the result that at least in low orders o f perturbation theory, they do not spoil the Drell-Yan picture. (b) The "Glauber" region: a ~ 13 ~ k2 /s, k± fixed (s -+ ~). It has recently been pointed out that this is an important region of phase space [5], moreover that contributions from two-loop graphs invalidate the usual Drell-Yan predictions. Below we shall study this in more detail. (c) The "infrared" region: a ~/3 ~ k ± / ~ , k± small (of the order of the infrared cut-off). Up to two years ago it was hoped that infrared divergences cancel in QCD in a way similar to the Bloch-Nordsieck mechanism in QED. We know now that this is not true, however it seems that the non-cancelling terms are suppressed by powers o f s [4] (or Q2). In this paper we shall be interested in the contributions from regions (a) and (b); specifically we shall integrate over all a and/3 keeping k± fixed. In order to be able to do explicit calculations we have chosen to work with massless scalar quarks, which are charged and coloured and described by the field X, and a neutral, colour singlet scalar hadron described by the field ¢. The hadronic wave function is then assumed to come from a XCX+Xinteraction term. X has the dimension of mass which has the effect that the wavefunction is soft, a physically desirable feature. Since we are dealing with massless quarks we have taken the hadron momentum, p, to be spacelike in order to guarantee the stability of the wavefunction. With this model we are able to study the problem of spectator quark interactions in detail, however since the results presented below seem to depend primarily on the analytic structure of the graphs (the ie terms in the propagators) we feel confident that they are more general than the model *l. The Sudakov variables for the outgoing quark momentum, P2, are
4 November 1982
F IDI(X ) = is 2 J -
*i It is for this reason that we do not worry about the unphysical nature of the spacelike hadron momentum. 106
k 2 + i e ) [ ( 1 - a ) ( p 2 - / 3 s ) - k 2 +ie]
X [(c~-x)(1 + / 3 ) s - k 2 +ie] -1 .
1 (5)
The factor i comes from the usual Feynman rules and the powers o f s come from the jacobian (d4k = ~s da × dfl d2kz) and the numerator (we have now changed variables k ~ P l - q - k from the labelling of fig. la). Now we consider the equivalent contribution from the graph shown in fig. lb to the Drell-Yan process quark + hadron -+ lepton pair + anything. This graph is very similar to that of fig. la and the integral to be calculated is
/By(O=-is2f (a~3s X [(1 - a -
da d/3 d2k± k 2 + ie)[(1 - a ) ( p 2 _/30 - k2 + ie]
o)(p 2 - p s - / 3 s ) -
× [(1 - o - a ) ( 1 - / 3 - n ) s - ( p
(P_L+kz)2 + i e ] - 1 a +k±) 2 +ie] -1 . (6)
The dominant contribution to these integrals comes from the small/3 region and it is a simple matter to show that IDI(r)
IDy(r) d2k±
= _4zr2i
f (Pl+k±)Z[k2-r(
1 - r)p2l
(7)
Thus the difference of the deep-inelastic and DrellYan amplitudes is purely imaginary. Since the contributions to the structure functions and Drell-Yan cross sections are equal to twice the real part of the interferences of the one-loop graphs of fig. 1 (together with the corresponding graphs involving "seagull" vertices) with the tree diagrams, we obtain no non-factorizing q
l
(4)
(or x +-+ r in the Drell-Yan case). After combining with the corresponding diagram involving a "seagull" vertex the integral to be evaluated is
(afls
X [(1 -- a - o)(p 2 - ; s --/30 -- (P± + k±) 2 + iel
P2 = ( o , p , p i } ((1 - x), p2/(1 - x ) s, P i }
da dfl d2k±
f
(o)
0 k,
P
~-
(b}
Fig. 1. One-loop graphs involving spectator interactions which contribute to (a) the structure function and (b) the Drell-Yan cross section quark + hadron -~ lepton pair + anything.
Volume 117B, number 1, 2
PHYSICS LETTERS
contribution from these one-loop graphs. Thus the Drell-Yan picture is not spoilt by '"collinear" or "Glauber" region contributions to one-loop graphs involving spectator interactions. On closer examination we find that the real parts of the two integrals in (5) and (6) are equal, whereas the imaginary parts have the opposite sign (but the same magnitude). The opposite sign comes from the ie's in the last propagator in (5) and (6), if we switch the sign of the ie in one of the two integrals then the imaginary parts become equal. We now come to the two-loop graphs. Bodwin et al., working in an axial gauge [5], suggest that the graph of fig. 2b has a contribution from the "Glauber" region which spoils the Drell-Yan factorization. We find however that (at least in the Feynman gauge) there is a precisely compensating contribution due to that of fig. 2a, which contributes to the structure function. The difference of the two corresponding integrals is again purely imaginary. We find that this happens for all the contributions except four! The first exception is the interference of the graphs of figs. 3a,b with those of figs. la,b, respectively. In the Drell-Yan case there is a contribution where we keep the imaginary parts of both figs. lb and 3b, with no corresponding contribution in the deep inelastic case since the diagram of fig. 3a is purely real (q2 < 0). The other three exceptions are the contributions of figs. 4, 5 and 6. In each case we Fred that the difference between the deep inelastic and Drell-Yan diagrams has a real part and has the correct scaling behaviour. Each of the four contributions is of the form ( g 4 / s ) ( 1 --
r){k2,k2s[r(1
× [¢(1 - ~ ) p 2 _ p ? ] - i
(o)
- r ) p 2 _ (p± + kl±)2 ] }-1 ,
(8)
4 November 1982
/ (o)
(b)
Fig. 3. Some one-loop graphs whose interference with those of fig. 1 gives a non-factorizing contribution for the Drell-Yan cross section (see text).
'k' {a)
{hi
Fig. 4. Two-loop graphs contributing to (a) the structure function and (b) the Drell-Yan cross section which are not related by the Drell Yan formula. with relative weights '
'
'
-- ~CFCA) , --7~CFCA , - ~ C F C A , from figs. 3, 4, 5 and 6, respectively. C F and C A are the eigenvahies of the quadratic Casimir operator in the fundamental and adjoint representations, respectively. All other contributions vanish when the deep inelastic contribution is compared to the Drell-Yan one and hence there are no nonfactorizing terms to this order!!! For completeness we would like to point out some of the things we have not yet calculated:
(b)
Fig. 2. Two-loop graphs contributing to (a) the structure function and (b) the DreU-Yan cross section which are related by the Drell-Yan formula (in the Feynman gauge).
(Q)
(N
Fig. 5. Two-loop graphs which are not related by the DreUYan formula. 107
Volume 117B, number 1, 2
(a)
PHYSICS LETTERS
(b)
Fig. 6. Two-loop graphs which are not related by the DrellYan formula. (a) We have not calculated the contribution from diagrams with real gluons. (b) We have restricted our transverse momenta to be much smaller than s and much larger than any infrared cut-off. We would like to extend the calculation to include all transverse momenta, however this is rather complicated at the two-loop level. (c) The higher order calculations in QCD [2] are performed with on-shell quarks and gluons. Perhaps the same calculations but now performed with one quark slightly off-shell will give a different result. At first sight this might be considered to be an attractive possibility since, for example, it is known that contributions from diagrams related to the Sudakov form factor depend on whether the external legs are on-shell or not. Landshoff and Stifling [7] have recently shown that this is precisely what happens in oneloop graphs when one calculates the K factor o f the D r e l l - Y a n process in h a d r o n - h a d r o n scattering. One can either calculate this factor with on-shell quarks and gluons and neglect spectator interactions (as is usually done) or one can calculate with both incoming partons off-shell, but then one has to include the spectator interactions. Both procedures give the same result. They find that the difference between the off-shell and onshell contributions from individual diagrams comes from the region o f small transverse m o m e n t u m (the "infrared" region) which is why we do not expect any
108
4 November 1982
real terms on the right hand side o f eq. (7) when considering only the "collinear" and "Glauber" regions. In the same way we would expect that at the two-loop level any contributions from diagrams not involving spectator interactions, which depend on whether the active quarks are on-shell or not would arise from the "infrared" regions and not from the phase space regions considered in this letter. However this still has to be checked. The results of this paper are reassuring in that we have not found any long distance contributions which cannot be absorbed into the parton distribution functions. It is now important to determine whether this is true to all orders, and if so, to understand why it is so in a more physical way, rather than relying on an apparently "miraculous" cancellation of terms. References
[1] S.D. DreU and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316; Ann. Phys. 66 (1971) 578. [2] G. Altarelii, R.K. Ellis and G. Martinelli, Nucl. Phys. B143 (1978) 521; B146 (1978) 544 (E); B157 (1979) 461; J. Kubar-Andr6 and F.E. Paige, Phys. Rev. D19 (1979) 221; B. Humpert and W.L. van Neerven, Phys. Lett. 84B (1979) 327;85B (1979) 293. [3] J. Badier et al., Phys. Lett. 89B (1979) 145; 96B (1980) 422; M.J. Corden et al., Phys. Lett. 96B (1980) 417. [4] R. Doria, J. Frenkel and J.C. Taylor, Nucl. Phys. B168 (1980) 93; C. Di'Lieto, S. Gendron, I.G. Halliday and C.T. Sachrajda, Nucl. Phys. B183 (1981) 223. [5] G.T. Bodwin, S.J. Brodsky and G.P. Lepage, SLAC preprint SLAC-PUB-2787 (1981). [6] C.T. Sachrajda and S. Yankielowicz, unpublished, reported in: J. Ellis and C.T. Sachrajda, Quantum chromodynamics and its applications in quarks and leptons, Proc. 1979 Cargbse Summer Institute, eds. M. L6vy et al. (Plenum, New York, 1980). [7] P.V. Landshoff and W.J. Stirling, Cambridge University preprint DAMTP 82/3 (1982).